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REVIEWFinal Exam

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Final Exam Information

DATE: THURSDAY, MAY 14

TIME: 10:00 AM - 12:00 noon

ROOM ASSIGNMENT:Butler-Carlton Hall

Room 315

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Final Exam Information• Comprehensive exam covers all topics

on syllabus.

• Equivalent to taking two regular exams.

• Format is similar to what you have seen on hourly exams throughout the semester.

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Strategy for Studying• Review topics:

– From the first half of the semester.– Topics that you are least comfortable with

• WORK homework problems and review problems and examples from the text.

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Test taking strategy

• Get to bed by midnight the night before.

• Eat a good breakfast.

• Work the problems you CAN and go back to the rest later.

• If you have time, take a two minute mental break and check over the whole exam once before turning in.

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Sign Convention Rules

• Always use a right handed coordinate system

• Designate the equation you are writing and indicate positive direction.

• Use right hand rule when evaluating moments.

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VectorsTypes of vectors:• Position vectors ( r )• Unit vectors ( u )• Vectors expressing a force ( F ) / moment ( M )

Vector operations:• Addition / subtraction• Dot product

– i*i + j*j + k*k– Scalar answer (NO < i, j, k > components; just magnitude)

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Particle Equilibrium - 2D & 3DStatic Equilibrium Equations:

FX FY (for 2D) FZ (for 3D)

Solving these problems:• Draw Free Body Diagram of point!

• Springs add additional equation FSP = ks

• Be able to do 3x3 matrix on calculator

F = 0 (for equilibrium problems)

OR

F = FR (for resultant force)

• 3D: Often better to use vector notation

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Moment Created by Forces

Static Equilibrium Equation: MPOINT

Solving these problems:

• Draw A PICTURE of the system!

• Scalar notation (M = F*d):

– Make sure moment arm is to force

– Watch sign convention and direction of moment created by each component of each force.

3D Moment vectors

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Moment Created by ForcesSolving these problems (…continued) :

• Vector notation ( M = r x F ):

– Watch direction of position vector ( r )

– r is always first in cross product.

• Moment about a line:

– Find moment about point ON line

– Then M = M uline

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Couple momentsCan sum moments for two forces creating a

couple at any point and the resulting moment

will be the same value.

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Equivalent Systems• Locate point of interest. If you have to find it, then use the problem to help

you find a preliminary “stand in” point.

• Sum forces (F = FR)

• Sum moments at point of interest (M = MR)

• If problem asks for resultant force ONLY, then find location (d) using equation MR = FR*d

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Distributed Loads

• Reduce to a single force by integrating the area under the load diagram.

• Find location of resultant force using the same method as for equivalent

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Equilibrium of a 2D Rigid Body

Equilibrium Equations for 2D:

FX FY Mpoint

Solving these problems: • Identify all forces and support reactions

• Draw Free Body Diagram of object!

• Sum forces in x and y direction.

• Sum moments at a point

• Don’t forget couple moments!

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Trusses: 2D Rigid Bodies

• Consist entirely of 2-force members.

• Direction of forces in members act along the direction of member (axially)

• Note axial direction of forces using (T) for tension and (C) for compression. DROP NEG. SIGN!!

• Analyze forces by cutting through internal members

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Analyzing Trusses• When cutting through members, this results in Free

Body Diagrams of:

– Single joint / pin connection

• Max of 2 unknowns FX   FY

– Section of structure (two or more joints)

• Max of 3 unknowns FX      FY  Mpoint

• Always assume unknown forces to be in tension (positive) negative answer indicates compression

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Frames and Machines• DO NOT consist entirely of 2-force members.

– Know how to identify 2-force members to help eliminate unknowns

– DO NOT cut through members, but take the structure apart at the connections.

• Direction of forces acting at connections is unknown, so make an assumption.

• Make sure forces on contacting members act in opposite directions.

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Analyzing Frames & Machines

• Write 3 equilibrium equations for each Free Body Diagram.

• You may have more than 3 unknowns on a particular diagram, so just solve what you can and move on …

• …Using the process of elimination to solve unknowns.

• The TOTAL number of unknowns cannot exceed the TOTAL number of equilibrium equations you can write.

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Comparison Table

Characteristics Trusses Frames & Machines

Consist of _______ members 2-force Multiforce

FBDsCut through members Take apart at joints

Assume direction of unknown is always tension

Doesn’t matter (except for cables)

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Internal Structural Loads

• Multi-force members carry – Moment load: bending (M) or torsional (T)

– Shear force (V)

– Normal force (N)

• To evaluate, cut through members

• Expose internal loads so they become external

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Sign Convention for Internal Loads

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Types of Problems: Evaluating Internal Loads

1. Cut through structure at specific point.

2. Cut through structure at arbitrary point.

3. Integrate load on beam (2x) and solve for

constants of integration (p.354-356).

4. Construct diagrams using “area method.”

(does not give equations, just a graph)

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Arbitrary Cut• Whenever there is a change in load, a discontinuity is created.

• Divide beam into regions determined by discontinuity.

• The beam shown below has three distinct regions bounded by

(0 ≤ x < 3) (3 ≤ x < 5) (5 ≤ x < 8)

• Set origin (x = 0) at left end of beam.

• Cannot evaluate internal shear EXACTLY at the point of an applied force. Must be just to the right or left of load.

Cut through structure at arbitrary point.

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Drawing V & M Diagrams• Plot directly under

beam

• x = 0 at left end

• Plot shear first

• Then plot moment

• Shear and moment are measured on y-axis

• Label each axis and graph including units

• Label all minimums and maximums, including points where shear crosses x-axis.

• SHOW calculations used to get values and draw graphs.

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Six Rules for V & M Diagrams

1. Concentrated force creates a jump in the shear diagram

Creates downward jump Creates upward jump

2. Change in shear equals area under load diagram

3. Slope of shear diagram equals value of distributed load

4. Change in moment equals area under shear diagram

5. Slope of moment diagram equals value of shear diagram.

6. Concentrated moment creates a jump in moment diagram

Creates downward jump Creates upward jump

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Dry Friction (Fs = s N)• We always evaluate the case for

impending motion

• # of unknowns = # of equil. eqns.

+ friction eqn.

• Dimensions must be given to check for tipping.

Mo to check for tipping

•If (x < l / 2) then block will slide.

•If (x > l / 2) then block will tip.

• Position of applied force influences how object responds

x = PhW

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remains stationary

impending motion

impending motion

• Uses the same friction equation as for dry friction – there are just more pieces.

• Be very careful when evaluating that you look at the direction of impending motion for each piece.

• Free Body Diagrams are absolutely essential.

Friction Wedges

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Belt FrictionT2 = T1e

• T2 is in the direction of impending motion

• T1 opposes direction of impending motion

= coefficient of friction between belt and surface of contact

= angle of contact between belt and surface

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Finding CentroidsCalculate as a weighted average:1. Compute the “moment” of each integral

element [weight, mass, volume, area, length] about an axis 2. Divide by total [weight, mass, volume, area, length]

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Using Single Integration1) DRAW an ‘element’ on the graph. 2) Label the centroid (x, y) 3) Label the point where the element intersects the curve (x, y) 4) Write down the general equation 5) Define each term according the problem statement 6) Determine limits of integration7) Integrate

~ ~

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Finding Centroids of Composite Shapes1) Divide the object into simple shapes. 2) Establish a coordinate axis system on the sketch 3) Label the centroid (x, y) of each simple shape4) Set up a table as shown below to calculate values 5) Subtract empty areas instead of adding them.

~ ~

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Resultant Force for Fluid on Surface

Flat surfaces

Solve for perpendicular resultant force

Fperp = z A

or

Fperp = (w1+w2)(1/2)(L)

Curved surfaces

Solve for vertical and horizontal components of resultant force and THEN find resultant.

Fv = (volume)

Fh = Fperp

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